Acceleration in One, Two, and Three Dimensions in Launched Roller Coasters

S PECIAL F EATURE: E XTREME P HYSICS

www.iop.org/journals/physed Acceleration in one, two, and three dimensions in launched roller coasters

Ann-Marie Pendrill

Department of Physics, G¨oteborg University, SE 412 96 G¨oteborg, Sweden

Abstract During a roller coaster ride, the body experiences acceleration in three dimensions. An accelerometer can measure and provide a graph of the forces on the body during different parts of a ride. To couple the experience of the body to pictures of the ride and an analysis of data can contribute to a deeper understanding of Newton’s laws. This article considers the physics of launched roller coasters. Measurements were performed with a three-dimensional co-moving accelerometer. An analysis is presented of the forces in the different ride elements of the Kanonen in G¨oteborg and the Speed Monster in Oslo, which both include loops and offer rich examples of force and acceleration in all dimensions.

Introduction The Stealth is the highest of the European 3, 2, 1 ... launch! The traditional lift hill, which launch coasters. After the launch, the train passes gives the initial potential energy for the ride, is the ‘top hat’ (figure 3) and then returns over a absent in some newly built roller coasters. Instead, camel back into a hairpin turn back into the station. the initial energy is provided in the form of a The Kanonen and Speed Monster roller horizontal launch, giving sufficient kinetic energy coasters both feature a loop and a screw during the to bring the train to the top of the first hill. ride. The accelerometer data and elevation profile From then on, the ride is characterized by the from these rides are shown in figures 4 and 5,and interchange between potential and kinetic energy, discussed in more detail below. in the same way as in traditional roller coasters. The first Intamin hydraulic launch coaster in One-dimensional horizontal motion Europe was Rita the Ride at Alton Towers, which In schools, the study of motion traditionally starts opened in April 2005, followed two weeks later by with non-motion, continuing with motion in one Kanonen at Liseberg in Göteborg (figure 1). The dimension. The traditional lift hill is an example Speed Monster at Tusenfryd in Oslo (figure 2)and of uniform rectilinear motion, where Newton’s the Stealth at Thorpe Park (figure 3) both opened first law applies. The launch is an example of in 2006. In 2007, similar launch coasters were accelerated motion in one dimension—as is the added to Heide-Park in Germany and PortAventura final brake. These situations can be useful as in Spain [1–3]. (See also the Roller Coaster Data illustrations to textbook presentations. In one Base at www.rcdb.com.) dimension, the measurement of the acceleration

Figure 1. The Kanonen roller coaster viewed from the side, showing the launch from the left into the ‘top hat’ on the right, as well as the shape of the clothoid loop.

Figure 2. Panorama of the Speed Monster. The launch is from the right into the Norwegian loop, which encircles the entrance escalator. (Photo: Jochen Peschel [1].) in the direction of motion gives full information nitrogen is compressed to a pressure of around about the motions if the initial speed is known. 300 bar. During launch, the gas is allowed to The variation of speed and distance with time is expand rapidly, sending the hydraulic oil through obtained by integration, which can be performed the motors, and energy is transferred to the numerically or analytically, after approximation of accelerating roller coaster. The technique is the acceleration time dependence. described in some detail by Peschel [3], who also presents an animation of the launch process. The pressure drops to about 250 bars, consistent The launch with the drop in horizontal acceleration during the Flags in the launch area enhance the sensation launch, seen from the graphs in figures 7 and 8. of motion during launch of the Speed Monster, Figures 7 and 8 shows the accelerometer as shown in figure 6. Horizontal launches of data for the Kanonen and Speed Monster rides. roller coasters have been used since the 1970s: The graphs also include speed and distance, for example, in the Revolution [2]whichisa obtained by numerical integration. From the Schwarzkopf ‘shuttle launch coaster’ [4], where graphs in figures 7 and 8, we can conclude that the energy is stored in a flywheel. Magnetic launch the force drops during the launch. This is natural techniques were introduced during the 1990s, with since the pressure of the nitrogen would drop as LIMs (linear induction motors) and LSMs (linear the gas expands, as discussed below. A fully synchronous motors). The compressed air launch loaded Kanonen train with four cars weighs about was introduced in 2002, followed by the hydraulic 8 tonnes. The Speed Monster train with three cars launch in 2002. The hydraulic launch was used is lighter, about 6 tonnes. These weights include to break a new altitude record in 2003 for the Top the mass of the sled used during acceleration. Thrill Dragster at Cedar Point, Ohio [2, 5]. Exercises for the reader. In the hydraulic launch, oil is pumped • What average power is needed to accelerate from a reservoir into storage cylinders filled the trains (in W and horsepower with nitrogen. The energy is built up as the (1 hp = 735 W))?

484 P HYSICS E DUCATION September 2008 Acceleration in one, two, and three dimensions in launched roller coasters

/g 2 vert a 0

0 5 10 15 20 25 30 35 40 45 t(s)

height (m) 0

0 5 10 15 20 25 30 35 t(s) Figure 5. Accelerometer and elevation data for the Speed Monster.

Figure 3. The 62 m high ‘top hat’ of the Stealth roller coaster at Thorpe Park.

4 Figure 6. The launch of the Speed Monster, with a side /g

tot view of the ‘Norwegian loop’. a 2 /g,

vert 0 a –2 • How high above the starting point can the 0 5 10 15 20 25 30 35 t(s) Kanonen and Speed Monster trains go after launch? 20

10 Acceleration measurements in three

height (m) 0 dimensions In roller coasters, as in everyday life, acceleration 0 5 10 15 20 25 30 35 t(s) is rarely restricted to one dimension. The forces Figure 4. Accelerometer and elevation data for required for the acceleration in a roller coaster are Kanonen. The green accelerometer curve shows evident throughout the body. What the body can magnitude of the ‘g-force’, whereas the blue curve shows only the vertical component. experience can also be measured with a co-moving sensor. Since the body moves in the gravitational ﬁeld, g, from the Earth, the additional force per • How does the power, P, vary during the mass unit required to obtain an acceleration, a,is launch? (Remember that power is force times (a − g). What is measured by an accelerometer is velocity, P = Fv.) thus in general not acceleration, but one or more

September 2008 P HYSICS E DUCATION 485 A-M Pendrill

30 ) 2 30 20 (m/s

x 20 a 10 10 (m/s),

0 x v 0 0 1 2 (m),

x 2 3 4 5 Figure 7. Horizontal acceleration (m/s2) for the t(s) Kanonen launch, together with velocity (m/s) and Figure 8. Launch of the Speed Monster: acceleration distance (m) obtained through numerical integration. (m/s2), velocity (m/s) and distance (s). Which graph is which? The drop in acceleration in figures 7 and 8 corresponds to a drop in force and thus to the drop in pressure during launch, which can be used to estimate the fraction of the maximum elements in a roller coaster, the lateral components possible work exerted by the gas during the Kanonen vanish if the curves are perfectly banked. and Speed Monster launches. A problem in measuring acceleration in three dimensions is to keep the sensor axis aligned components of this vector. Since the gravitational with the body axis. When the sensor is kept acceleration is used as a reference, it is natural to safe in a vest on the body, the z-axis tends give results in terms of the ratio (a − g)/g.This to slope slightly backwards and sometimes also expression can be taken as a vector definition of sideways. A mathematically simple option is to |a − g| the ‘g-force’. use the magnitude of the vector , possibly incorporating the sign from the dominating The accelerometer data in this paper were vertical component to maintain ‘negative g’ obtained using a wireless dynamic sensor system readings. The Kanonen data in figure 4 show a from Vernier. This system also measures the comparison of the total g-force and the vertical air pressure and converts the barometer data to component. provide indications of altitude during the ride. When the other coordinates are also of Through Bernoulli’s principle, the altitude data interest, as for launch, break and roll, it is are influenced by speed, thus leading to an necessary to perform a coordinate transformation. overestimate of altitude for high speeds. (This can The data in this paper were transformed by rotating be seen, for example, around launch in the graphs the axes so that the data have only a vertical in figures 4 and 5.) component before the ride starts, and assuming that the sensor orientation relative to the track is Coordinate system for amusement ride fixed. acceleration data The experience of the body depends on the Two-dimensional motion in loops orientation. A natural coordinate system to Both the Kanonen and the Speed Monster include describe the experience follows the moving body, loops, where the train moves essentially in two thus changing direction throughout the ride, and dimensions. The photographs in figures 1 and 9 this is also the coordinate system used by the show that neither the loop in the Kanonen nor sensor to record the motion. Here, we define the in the Speed Monster is a perfect circle. In a positive z-axis to be the ‘vertical’ axis directed circular loop, weightlessness at the top would be along the spine towards the head of the rider. The accompanied by 6g at the bottom of the loop positive x-axis points to the front of the rider—in (neglecting energy losses and the length of the most roller coaster rides, including these ones, the train). To reduce the load on the body, the shape x-axis coincides with the direction of motion. The of the track has a larger radius of curvature at y-axis gives the direction of the ‘lateral’ g-force. the bottom. This can be achieved in different In a right-handed system it will point out to the ways, as discussed in more detail in [6, 7]. The left of the rider. Apart from launch and brake, the Kanonen loop is a classic ‘clothoid loop’, which longitudinal component should vanish if friction was introduced by Werner Stengel in 1976 in the and train length are neglected. Except for screw roller coaster revolution [6].

486 P HYSICS E DUCATION September 2008 Acceleration in one, two, and three dimensions in launched roller coasters

Figure 9. The ‘Norwegian loop’ of the Speed Monster roller coaster encircles the roller coaster entrance to the Figure 10. The large corkscrew of the Speed Monster. park, making a very large loop possible. In view of the Technically, only the last hill is considered as a short Speed Monster train, the ratio between train corkscrew element. The track twists so that the train length and loop radius thus becomes unusually small in runs on top of the track in the first two coils. Only the this case. last coil leads to an inversion of the rider. The train position in the photo corresponds to t = 33 s in the graph in figure 5. In traditional roller coaster loops, the train enters the loop from below. The Speed Monster train instead enters the loop from above. This accompanied by a perpendicular motion along the feature, conceived by project director Morten cylinder axis. In the photograph of the corkscrew Bjerke at Tusenfryd, makes the Speed Monster in figure 2, the track seems a bit flattened at the top. loop unique. Is is classified as a ‘Norwegian At the same time, the track twists, so the heartline loop’ in the Roller Coaster Data Base (www.rcdb. of the rider moves more along the cylindrical com). It gives the rider two inversions, during both shape. Let L denote the distance between the coils entrance to and exit from the loop. along the axis. For the train to move a full coil, it The Kanonen train passes the highest point at then moves a distance 2π R around the circle and time 15 s in the data series in figure 4, showing L along the axis. The velocity component along essential weightlessness at the top and close to the cylinder axis is unchanged during the motion. 4g during entrance to and exit from the loop. The angle of the track to the axis is given by Similarly, the Speed Monster rider is essentially weightless at the entrance and exit from the loop tan α = 2π R/L. (at 10 s and 15 s, respectively, in figure 5), while experiencing close to 4.5g at the bottom. Exercise. Comparing the loop shapes, we see that, • Show that a train moving with speed v along whereas the traditional loops are somewhat a corkscrew track leads to a centripetal narrower than a circle, the larger curvature at the acceleration with magnitude bottom of the Norwegian loop leads instead to a slightly wider shape. (v sin α)2 v2 1 a = = . c R R 1 + L2/4π 2 R2 Three-dimensional motion in corkscrews • Show that the difference between the g-force The picture of the Speed Monster launch (figure 6) at the top and bottom of the corkscrew is also shows the large Norwegian loop from the given by side. All of the loop is nearly in the same plane. Separating the coils by a larger distance would lead 4 2g + 4g sin2 α = g 2 + . to a corkscrew, such as in the Speed Monster, as 1 + L2/4π 2 R2 seen in figures 2 and 10. A corkscrew can, as a first approximation, In the formula above, the first 2g arise due be described in cylindrical coordinates, where to the different direction of the body relative to the circular motion with a radius R is then gravity, when the rider stays on the inside of

September 2008 P HYSICS E DUCATION 487 A-M Pendrill

Kanonen, Heartline Roll

2 total /g a 0

lateral vertical –2 25 26 27 28 29 30 31 32 33 t(s) Figure 12. Accelerometer data for the ‘heartline roll’. The vertical (red) and lateral (green) components are shown together with the total g-force (black) on the Figure 11. The photograph shows the Kanonen train on body. Since the body moves with essentially constant the way back through the loop into the heartline roll, velocity, the total force from the train on the body is where the centre of mass of the rider moves essentially mg, counteracting the force of gravity throughout the along a straight line. roll. However, the direction is changed relative to the rotating coordinate system of the body and of the accelerometer. the screw and is upside down at the top. This obviously does not apply in situations, such as ‘heartline roll’. The body’s centre of mass moves figure 12, where the train has twisted around to the with nearly constant velocity. What forces act on top of the track in the highest point. Corrections the body? Figure 12 shows the accelerometer data may also arise due to the motion of the train around for this part of the tour. the track. Any difference in radius of curvature It is tempting to believe that measurement between the high and low points also leads to a with a three-dimensional accelerometer gives a change in g-force difference to what is expected complete description of the motion, which can be from these formulae. used to recreate the shape of the track. However, the accelerometer data between 27 and 30 s in Speed Monster corkscrew figure 12 could be obtained without rotation by moving up–down and left–right, some 20 m in The corkscrew in the Speed Monster is quite each direction (although the altitude profile does, stretched, making good use of the available space, indeed, show that this was not the case). as seen from the panorama picture in figure 2.In Newton’s first law tells us that a body remains the Kanonen ride, the corkscrew is stretched to the in uniform rectilinear motion unless acted on by point where the riders move along a straight line unbalanced forces. However, when the ‘body’ while the track twists around them, giving a ratio in Newton’s laws is our own it is clear that R/L close to zero. the direction of the forces relative to the body As an exercise, estimate R/L for the Speed matters: we are not point-like particles. A ‘motion Monster corkscrew from figure 2. Use this tracker’ needs also to measure rotation around ratio to estimate the difference in g-force for the the three axes to get a complete description of different parts of the ride. Does your result agree the motion [9]. Nevertheless, three-dimensional with the accelerometer data in figure 5,where accelerometer data provide much material for the corkscrew spans the period of about 10 s, analysing familiar motions. starting at t = 28 s? Are there any deviations from expectations that would prompt you to make Personal experiences additional observations or measurements in the park? The wireless sensor system The WDSS system is extremely simple to use. The heartline roll of Kanonen Once set up from a computer, it can be used by a large number of students to collect data over On the way back to the station, the Kanonen train a whole afternoon. It is, however, important to performs a show-off passage over guests in the keep notes of what rides have been studied, since ◦ queue (figure 11). The track turns about 270 in a no additional information is stored on the sensor.

488 P HYSICS E DUCATION September 2008 Acceleration in one, two, and three dimensions in launched roller coasters Should the memory fill up, the data are quickly are, of course, during off-season, when you do not transferred to a laptop, and the sensor is again have to wait more than a few minutes to board the ready for additional measurements. train. The measuring vest, although not particularly Rita—Queen of Speed, Kanonen and the aesthetically pleasing, seems to convince ride Speed Monster all have slower speeds and lower attendants that the wearer is serious. Most hills than the Stealth. Rita—Queen of Speed importantly, the vest keeps the sensor from falling reaches a higher speed (98 km h−1) than the out during the ride: safety concerns must always Scandinavian launch coasters, but the ride heights come first. A disadvantage is, however, that it is at Alton Towers are limited by the tree tops. The difficult to keep the coordinate axes aligned [10]1, speed gained from the launch is instead used in a but in most cases this can be dealt with afterwards. helix with an extended period of relatively strong One-dimensional accelerometer data are suf- g-forces. The whole 640 m tour in Rita the Ride ficient to obtain velocity and position rectilinear lasts 25 s, which may seem a bit short after a motion, at least in principle. For a complete de- long time in the queue. Alton Towers has more scription of three-dimensional motion, accelerom- rewarding roller coaster rides! eter data for the three axes must be complemented The Kanonen and the Speed Monster both by rotational data around all axes, as discussed by turn the rider upside down a few times during Pendrill and Rödjeg˚ard [9], in connection with the the ride, in loops and screws, discussed above. analysis of motion tracker data for a roller coaster. Although the inversions could be captured by a Still, the simplicity of use for the WDSS sensor rotational sensor, the visual experience could not. makes it a useful tool, bringing the amusement The Kanonen launch goes across a small river, park experience to the classroom. giving the riders the impression of falling into the water after the top hat. The Speed Monster The roller coasters has a most spectacular track layout, encircling the Although a three-dimensional accelerometer records entrance escalators. It runs on a hillside, and the time series of forces acting on the body, it can brings the rider through the terrain, close to the obviously not capture the whole experience. tree tops. The Kanonen track is woven back and Part of the experience is the build-up of forth, making maximum use of a small available expectations during the time in the queue. The area. Its complicated structure is more difficult to Stealth queue at Thorpe offers TV screens with memorize, which possibly brings more surprises its own disc jockey. During my one-hour wait to to the rider. The Speed Monster makes use of the get on (August 2006), people in the queue were natural drops to bring the train considerably below dancing to the music, and in general having a the starting point, thereby increasing the maximum good time. There was also the occasional speaker speed. The ride is only about 2 s longer than message telling NN to get ‘back to the entrance the Kanonen ride, as seen from the accelerometer where mum has got a FastTrack ticket for you’. data. However, the difference feels larger, possibly Just before entering, the riders are brought in close because the Speed Monster track is more than 50% view of parts of the launch technology. The queue longer: 690 m compared to 440 m. (The Stealth also lets you look up towards the 62 m high top tour is even shorter: 400 m.) The longer track hat (figure 3), and I have to confess that it was also accounts for the smoother ride, where more the first time in many years that I had the feeling distance is allowed for the different elements [8]. ‘am I really going to go on that ride?’. But, yes, Which coaster is the ‘best’? To some extent I did, and even got a FastTrack ticket for a second this depends on your personal preferences. The ride. The long period of acceleration followed by results from annual voting by riders can be found weightlessness is quite a strong experience. at BestRollerCoasterPoll.com. Both the Kanonen and the Speed Monster offer good views of different parts of the ride as Lessons in the amusement park or roller coasters the queue moves on. The best queuing experiences in the classroom?

1 Figure 3 of [10] shows the spillover from the vertical Is it best to have physics lessons in the amusement accelerometer reading, which results when the coordinate axes park or to have lessons in school about physics are not correctly aligned. in amusement rides? Even without easy access

September 2008 P HYSICS E DUCATION 489 A-M Pendrill to an amusement park, most students are likely the top of the first coil (figure 6, at about 28 s in to have been on the rides and can relate the the data shown in figure 5). Measurements are not experience of their body to the physics description only about numbers, but about questions, answers of the rides. Swings in a nearby playground and insight. are a good way to introduce amusement park physics [11]. Discussions of forces in the rides are likely to change students’ ways of thinking Acknowledgments during future park visits, as many students have First, I would like to express my appreciation reported. As with all field trips [12], the learning to roller coaster designer Werner Stengel for outcome from an amusement park visit depends kindly sharing part of his knowledge about various to a large extent on the preparation. Several aspects of roller coasters, including loop shapes. Internet sites provide material for preparing I would also like to thank Jochen Peschel from amusement park visits (e.g. [13, 14]). Some Coasters and More for permission to use the parks, including Alton Towers and Thorpe Park, photograph in figure 2, and for interesting e-mail offer educational programmes for visiting school classes. Thorpe Park also quotes education correspondence. Finally, I would like to thank secretary Alan Johnson: ‘Learning outside the the helpful people at Liseberg and Tusenfryd, in classroom should be at the heart of schools’ particular Ulf Johansson and Morten Bjerke, for curriculums and ethos.’ practical help and for stimulating discussions. Measurements in the park can easily over- Received 2 January 2008, in final form 6 February 2008 shadow analysis, which is left for later. Back doi:10.1088/0031-9120/43/5/003 in the classroom, the rides are no longer at hand for investigating questions arising from the data. The balance between measurement and analysis is References worth careful consideration. The analysis of mea- [1] Peschel J 2006 Speed monster—Powerrausch am surement data also takes somewhat different forms Felshang Coasters and More www. depending on what data can be obtained from the coastersandmore.de/rides/speedmonster/ park. Drawings are usually secret, as required by speedmonster.shtml the agreements between parks and designers. The [2] Marden D 2003 3, 2, 1, launch! Fun World Mag. www.iaapa.org/industry/funworld/2003/Jul03/ length of a roller coaster train can, however, usu- Features/3 2 1 Launch!/3 2 1 Launch!.html ally be obtained. (If not, it can be estimated by [3] Peschel J 2007 Xcelerator—Intamin’s accelerator measuring the width of the gates in the boarding coaster premiere Coasters and More www. queues.) It can provide a length scale for analy- coastersandmore.de/rides/xcel/xcelerator. sis of different elements of the roller coaster from shtml, see also www.coastersandmore.de/rides/ kanonen/kanonen eng.shtml photographs or video clips. The length, combined [4] Schwarzkopf A 1979 Amusement ride with with the time of passage at a given point, gives a vertical track loop US Patent Specification speed measurement. Comparing timing from stop 4165695 (DE2703833) watches of a number of students’ mobile phones [5] Higgins A 2003 The coaster with the moster provides good material for discussions of measure- Machine Design www.machinedesign.com/ ASP/strArticleID/55720/strSite/MDSite/ ment uncertainty. Sometimes the track layout also viewSelectedArticle.asp makes it possible to estimate energy losses from [6] Schützmannsky K 2001 Roller Coaster—Der measurement of the time of passage. Achterbahn-Designer Werner Stengel I find that every time I get new data from a Ingenieurbüro Stengel www.rcstengel.com [7] Pendrill A-M 2005 Roller coaster loop shapes ride, they give rise to questions, and an urge to go Phys. Educ. 40 517–21 back and check. Now, I would like to go back to [8] Stengel W 2007 Private communication Tusenfryd and take a good look at the corkscrew [9] Pendrill A-M and Rödjeg˚ard H 2005 A roller to see if what looks like an extra large radius of coaster viewed through motion tracker data curvature at the bottom of one of the coils can Phys. Educ. 40 522–6 [10] Butlin C A 2006 Flying high with sensor system explain the dip in g-force around 35 s in figure 5; Phys. Educ. 41 577–9 and I would need to ride it again to feel that dip. [11] Pendrill A-M and Williams G 2005 Swings and I would also like to feel the ‘negative g-force’ at slides Phys. Educ. 40 527–33

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[12] Rennie L R and McClafferty T P 1996 Science Ann-Marie Pendrill is professor in centers and science learning Stud. Sci. Educ. physics at G¨oteborg University, with a 27 53 background in computational atomic [13] Bakken C 2007 Physics/Science/Math physics. Her teaching involves Days@Great America http://physicsday.org engineering, physics and teacher [14] Pendrill A-M 2007 Science in the Liseberg programmes and she is involved with amusement park http://physics.gu.se/ different forms of informal learning, LISEBERG/ including amusement park physics.

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